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This paper provides a theoretical study and calculation of the specific detectivity-
D* limit of photovoltaic (PV) mid-wave infrared (MWIR) PbSe n
^{+}-p junction detectors operating at both room temperature and TE-cooled temperature. For a typical PbSe p-type doping concentration of 2 × 10
^{17} cm
^{-3} and with high quantum efficiency, the
D* limits of a photovoltaic PbSe n
^{+}-p junction detector are shown to be 2.8 × 10
^{10} HZ
^{1/2}/W and 3.7 × 1010 HZ1/2/W at 300 K and 240 K, with cut-off wavelength of 4.5 μm and 5.0 μm, respectively. It is almost one magnitude higher than the current practical MWIR PV detector. Above 244 K, the detector is Johnson noise limited, and below 191 K the detector reaches background limited infrared photodetector (BLIP)
D*. With optimization of carrier concentration,
D* and BLIP temperature could be further increased.

The MWIR light detection has widespread applications in the fields of health monitoring, environmental protection, defense and national security, as well as space exploration and other fields. Existing technologies with high sensitivity are mainly based on semiconductor photo-detectors. In the past half-century, many semiconductor material systems have been intensively studied, and significant progress has been made [

It is well known that Auger coefficient in IV-VI semiconductors [^{10} cm・Hz^{1/2}/W and 4.2 × 10^{10} cm・Hz^{1/2}/W at ~3.8 µm, with and without antireflective coating [

It is worth nothing that photovoltaic (PV) detectors offer advantages over their PC detector counterparts, such as low power consumption, high pixel density, and small pixel size. In addition, a PV detector FPA operating at low bias with no or low flicker noise eases the design of ROIC which further reduces cost. Although the D* of the IV-VI Pb-salt semiconductor junction at a low temperature has been investigated via experimental and theoretical research, the performance limit of PbSe at high temperature has not been well investigated. In this paper, we investigate the performance limit of a PbSe n^{+}-p junction detector operating at room temperature and TE-cooled temperature.

Specific detectivity, or D*, for a photodetector is an important figure of merit and is determined by [

D * = A Δ f N E P (1)

where A is the active area of the detector and NEP is the noise equivalent power (the power needed to generate the signal current which is equal to the noise current)

N E P = Φ e i s i g / i n (2)

Here Φ e [

In a junction device, the fluctuation of diffusion rates in the neutral region and the generation-recombination (g-r) fluctuation in both depletion region and quasi-neutral region are indistinguishable. All of them give rise to the shot noise which has the form of [

i n 2 = 2 q ( I D + 2 I S ) Δ f (3)

where I D is the diode current, I S is the dark reverse-bias saturation current of the device and Δ f is the electrical bandwidth.

I D = I S × ( e q V β k T − 1 ) (4)

β is the ideality factor which determines how the actual diode deviates from the ideal diode. In most of our later calculations, we assume the ideal diode case where β = 1 .

For the diode under the illumination of photon density flux Φ b , which generates the photon current I p h = q η A Φ b , the total noise contribution is:

I n 2 = 2 q ( q η A Φ b + k T q R 0 e q V k T + k T q R 0 ) Δ f (5)

where R_{0} is the zero-voltage resistance which is defined by

R 0 = ( ∂ I D ∂ V ) V = 0 − 1 = k T q I S (6)

Under the zero-voltage case, the expression of shot noise in (3) has the same form as the Johnson-noise associated with the R_{0} of the junction, thus it is often called Johnson noise.

Substituting (1) and (2) to (5), the expression for the detectivity of the device is

D * = q η λ h c [ 4 k T R 0 A + 2 q 2 η Φ b ] − 1 / 2 (7)

In Equation (7), η is the quantum efficiency, λ is the wavelength of the incident radiation, q is the charge of the carrier, h is Planck’s constant, c is the speed of light, k is Boltzmann’s constant, R_{0} is the diode incremental resistance at 0 V, A is the detector sensitive area, T is the temperature of the detector in Kelvin, and Φ b is the photon flux incident on the detector from its surroundings. It is clear that both R_{0}A and η are key factors on the specific detectivity D*. Therefore, the resistance-area (R_{0}A) product at zero bias needs to be calculated to determine D* based on Equation (6).

The total dark current density flowing through the junction can be found with the following equation:

J s = I s A = J D F + J G R + J T + J L (8)

In Equation (7), J_{DF} is the diffusion current density, J_{GR} is generation-recombination current density that is often dominated in the depletion layer, J_{T} is tunneling current density and J_{L} is leakage current density. J_{L} may be due to bulk as well as surface defects of the material. When suitable diode technology and construction are used, the contribution of the J_{L} component is negligible.

The maximum R_{0}A is obtained from the one-sided abrupt junction [^{+}-p junction will dominant the diffusion current. The R_{0}A product determined by the diffusion current in the case of radiative recombination and Auger recombination is

( R 0 A ) D F = ( k T ) 1 / 2 q 3 / 2 n i 2 N A ( τ e μ ) 1 / 2 (9)

where n_{i} is the intrinsic carrier concentration, N_{A}(N_{D}) is the concentration of donors (acceptors), τ_{h}(τ_{e}) is the minority carrier lifetime of holes (electrons) and μ_{h}(μ_{e}) is the minority carrier mobility.

The resistance-area product (R_{0}A) of PbSe photo-voltaic detectors under conditions of zero bias is calculated for one-side abrupt junction. Highly doped n-side homojunction (n^{+}-p) is discussed in this paper. At room temperature, the effective R_{0}A product on the PbSe n^{+}-p junction can be expressed by:

( R 0 A ) − 1 = ( R 0 A ) D F − 1 + ( R 0 A ) T − 1 + ( R 0 A ) S R H − 1 = ( R 0 A ) R A − 1 + ( R 0 A ) A u g e r − 1 + ( R 0 A ) T − 1 + ( R 0 A ) S R H − 1 (10)

where ( R 0 A ) R A is the R_{0}A product due to radiative recombination, ( R 0 A ) A u g e r is the R_{0}A product due to Auger recombination, and ( R 0 A ) S R H is the R_{0}A product due to Shockley-Read-Hall recombination. Different R_{0}A products due to each recombination mechanism will be discussed in the following section.

The R_{0}A product determined by the diffusion current in the case where radiative recombination has the relationship on the lifetime τ r − 1 . The radiative lifetime τ r − 1 for PbSe follows as [

G R = 10 − 15 n r ( k T ) − 3 2 ( 2 + 1 K ) − 3 2 ( m * m ) − 5 2 ( K ) − 1 2 E g 2 cm 3 / sec , K = m l * / m t * (11)

τ r − 1 = G R ( n 0 + p 0 ) (12)

n 0 + p 0 = 2 ( N D 2 4 + n i 2 ) 1 / 2 (13)

The radiative lifetime is inversely proportional to the total concentration of free carriers under all circumstances. The radiative lifetime decreases when the doping concentration increases.

G_{R} is the capture probability for radiative recombination. The band gap of a semiconductor E_{g} is expressed in electron-volts. n_{r} is the index of refraction, and m * = [ 1 / 3 ( 2 / m t + 1 / m l ) ] − 1 is the density-of-states electron and hole effective masses. n_{i} is intrinsic carrier concentration. n_{0} and p_{0} represent the equilibrium carrier concentrations. According to Equation (14), the (R_{0}A)_{RA} product can be found after the radiative recombination lifetime is calculated.

( R 0 A ) R A = ( k T ) 1 / 2 q 3 / 2 n i 2 N A ( τ r μ ) 1 / 2 (14)

^{17} and 10^{18}, the radiative recombination life time of PbSe is between 10^{−}^{7} s and 10^{−}^{8} s at 300 K.

The R_{0}A product determined by the diffusion current in the case of Auger recombination has the relationship on the lifetime τ A − 1 . The PbSe Auger’s recombination coefficient C_{A} is given by [

C A = 3 q 4 ( 2 π ) 5 / 2 ( k B T ) 1 / 2 E g − 7 / 2 ћ 3 ( 16 π ε 0 ε ∞ ) 2 m l ∗ 1 / 2 m t ∗ 3 / 2 × exp [ − E g 2 k B T ( m l * m t * ) − 1 ] (15)

where m l * and m t * are the longitudinal and transverse effective mass, ε ∞ is high frequency dielectric constant, and ε 0 is vacuum permittivity. During the Auger recombination process, the carrier lifetime is defined as:

τ A = [ C A ( N A 2 + 2 n i 2 ) ] − 1 (16)

^{17}^{ }

and 10^{18}, the Auger recombination life time of PbSe is between 10^{−}^{7} s and 10^{−}^{8} s at 300 K. According to Equation (17), the (R_{0}A)_{Auger} product can be derived when the Auger recombination lifetime is calculated.

( R 0 A ) A u g e r = ( k T ) 1 / 2 q 3 / 2 n i 2 N A ( τ A μ ) 1 / 2 (17)

The (R_{0}A)_{T} determined by tunneling is given by

( R 0 A ) T = 4 ∗ π 3 ћ 2 ( ε s ε 0 ) 1 2 exp [ π ( m x ε s ε 0 ) 1 2 E g 2 3 2 q ћ N D 1 2 ] / ( q 3 2 N A m y m z m x ) (18)

Tunneling simulation of PbSe depends on the crystal orientation due to the difference in effective masses. The four effective masses to consider are: m e l (conduction band, longitudinal), m e t (conduction band, transverse), m h l (transverse band, longitudinal) and m h t (transverse band, transverse). m 0 is the electron resting mass [

m e l = [ 11.4 × 0.145 E g + 2.9 ] − 1 × m 0

m e t = [ 20.7 × 0.145 E g + 4.3 ] − 1 × m 0

m h l = [ 11.4 × 0.145 E g + 3.3 ] − 1 × m 0

m h t = [ 20.7 × 0.145 E g + 8.7 ] − 1 × m 0

For PbSe [

In order to compare Auger, radiative and tunneling recombination, R_{0}A production on those three recombination mechanisms versus different temperatures is analyzed. In the following calculation N A = 2 × 10 17 cm − 3 , a typical PbSe hole concentration is used. To satisfy n^{+}-p junction, N_{D}_{ }is chosen to be 1 × 10^{18} cm^{−}^{3}, which is five times higher than N_{A}. Throughout this paper, we use N_{D} ≥ 5N_{A} for our n^{+}-p junction calculation. Looking at _{0}A)_{T} increases significantly for the PbSe [_{0}A)_{T} can be ignored. This is mainly because the energy bang gap of PbSe increases with temperature. When

m_{x} | m_{y} | m_{z} |
---|---|---|

m_{t} | 3 m l m t 2 m l + m t | 2 m l + m t 3 |

the temperature is around 160 K, (R_{0}A)_{T} is comparable to the Auger R_{0}A and radiative R_{0}A. For even lower temperatures, (R_{0}A)_{T} becomes dominant. At temperatures above 180K K (R_{0}A)_{T} is at least five times higher than the other mechanisms and thus (R_{0}A)_{T} can be neglected relative to the total R_{0}A product.

To keep the same N_{A} and increase N_{D} doping concentrations, _{0}A)_{T} cannot be neglected even at room temperature. At about 275 K, (R_{0}A)_{T} is five time lower than the other mechanisms and at higher than 275 K tunneling needs to be considered in the total R_{0}A product. Comparing _{0}A, and therefore, the total device performance.

The carrier generation-recombination mechanisms in detector devices are distinguished as Auger, radiative, tunneling, and Shockley-Read-Hall’s (SRH) generation-recombination mechanisms. Auger, radiative, and tunneling mechanisms are determined by energy band structures. However, SRH’s mechanism is determined by the material quality. In Equation (4), the depletion generation and recombination R_{0}A term is largely dominated by the lifetime associated with SRH centers. However, the value of this lifetime is extremely uncertain as has been noted in previous studies [^{−8}s^{25}. Thus, using this assumed SRH lifetime, higher temperature operations produce high values for (R_{0}A)_{GR} which may be neglected in our case. However, SRH recombination may need to be considered with a shorter SRH lifetime.

Utilizing the R_{0}A expressions shown in the above sections, calculations of R_{0}A for Auger, radiative, and tunneling were described in _{0}A dependence on carrier concentration reveals the contribution of each mechanism, revealing an optimized condition for the maximum R_{0}A limit. Three temperatures of interest are used. 300 K, 240 K, and 160 K represent temperatures of an uncooled, cut-off wavelength of 5 μm, and that the tunneling effect needs to be considered.

μ = μ 0 T − 2.65 (19)

And the expression of energy gap is given as [

E g ( T ) = 125 + 400 + 0.256 × T 2 (20)

At 300 K, the total effective R_{0}A product is around 1.3 × 10^{−}^{4} ohm・m^{2}. At 240 K, the total effective R_{0}A product is around 1.6 × 10^{−}^{4} ohm・m^{2}. When temperature is higher, the total effective R_{0}A product becomes smaller. According to Equation (10), _{0}A product on the

Eg(eV) | 0.278 eV |
---|---|

P_{l}(eVcm) | 3.9 × 10^{−8}* |

P_{t}(eVcm) | 2.9 × 10^{−8}*^{ } |

μ(cm^{2}/Vs) | 200 |

ε_{s}_{ } | 203 |

ε_{∞ } | 22.9 |

diffusion current in the case of Auger recombination and radiative recombination at a different temperature. At 160 K, tunneling R_{0}A value is comparable to the Auger and radiative R_{0}A value.

_{0}A limitation at 160 K when the n^{+} doping concentration is N D = 1 × 10 18 cm − 3 . In this case, ( R 0 A ) T is comparable with ( R 0 A ) R A − 1 and ( R 0 A ) A u g e r − 1 . When the tunneling effect on the total R_{0}A cannot be ignored, there exists an optimized carrier concentration for the largest R_{0}A. In this case, the largest R_{0}A occurs at N A = 1.8 × 10 17 cm − 3 . When the n side carrier concentration was increased from 1 × 10 18 cm − 3 to 5 × 10 18 cm − 3 , tunneling could not be ignored at 300 K, as shown in _{0}A in _{A}) gives a larger total R_{0}A. This is because a larger N_{A} leads to a smaller minority carrier concentration, which in turn reduces the saturation current. However, for a n^{+}-p junction n-doping concentration has to be increased, as p-doping concentration increases, which will increase the tunneling effect. In our simulations, the maximum R_{0}A limit at 240 K (^{+}-p junction where N D = 1 × 10 18 cm − 3 and N A = 2 × 10 17 cm − 3 . The recombination mechanisms in n^{+} side are not taken into consideration in this paper. Recombination in n^{+} side poses a further limitation of n^{+} concentration and thus p-doping concentration for optimized D*. Assuming a wide bandgap N^{+}-doping layer with negligible Auger and Radiative recombination rate, and negligible tunneling effect, then higher p-type doping concentration could be used to further increase the R_{0}A.

As can be seen from Equation (6), the two main mechanisms contributing to the homojunction device are the thermal noise and the background photon shot noise. By using calculated R_{0}A when N D = 1 × 10 18 cm − 3 and N A = 2 × 10 17 cm − 3 , the temperature dependence current noise of these two sources is described in

From

D * = q η λ h c [ 4 k T R o A ] − 1 / 2 for T > 244 K (21)

D * ≡ D B L I P ∗ = q η λ h c [ 2 q 2 η Φ b ] − 1 / 2 for T < 191 K (22)

To find the theoretical limit for PbSe photovoltaic detector D* at 300 K, we apply the R_{0}A product from the above conditions with Equation (23) because photon shot noise can be negligible at room temperature.

D * = q η λ h c [ 4 k T R o A ] − 1 / 2 (23)

At 240 K, to calculate the theoretical limit for PbSe photovoltaic detector D* following by:

D * = q η λ h c [ 4 k T R o A + 2 q 2 η Φ b ] − 1 / 2 , Φ b = 1.23 × 10 16 cm − 2 ⋅ sec − 1 (24)

Theoretically, the quantum efficiency should be calculated by the number of electron-hole pairs generated per incident photon. In order to get the D* limitation, the detector quantum efficiency is simplified to a constant for a different cutoff wavelength. The D* of the PbSe junction is simulated for a quantum efficiency of 75% and 100%, λ is the wavelength of the incident radiation, q is the charge of the carrier, h is Planck’s constant, c is the speed of light, k is Boltzmann’s constant, R_{0} is the diode incremental resistance at 0 V, A is the detector sensitive area, T is the temperature of the detector in Kelvin. With cutoff wavelength λ = 5 μm at T = 240 K, the photon flux Φ b equals to 1.23 × 10^{16} cm^{−2} sec^{−1}. ^{+}-p junction detector is 2.8 × 10^{10} HZ^{1/2}/W at 4.5 μm. At 240 K, the D* theoretical limit is 3.7 × 10^{10} HZ^{1/2}/W at the cut-off wavelength of 5 μm.

Due to reabsorption in the crystal, the effect of radiative recombination may be reduced and carrier radiative life time becomes larger [

At 300 K, the limitation of D* now increases from 2.8 × 10^{10} HZ^{1/2}/W to 3.9 × 10^{10} HZ^{1/2}/W at 4.5 μm when radiative recombination is ignored.

In this n^{+}-p homojunction junction model, the n^{+} PbSe was not considered for its impact on diffusion R_{0}A product. As discussed in previous sections, high carrier concentration will lead to high Auger and radiative recombination and thus reduced R_{0}A product. To achieve the calculated D* in this paper, a heterojunction structure with a wide bandgap N^{+} doped layer with much reduced Auger and radiative recombination is needed.

According to _{0}A value increases with increasing N_{A}. However, to keep the device n^{+}-p junction N_{D} needs to be increased with N_{A}, which increases the tunneling effect. Therefore, there exists an optimized N_{A} which provides the maximum R_{0}A at a given temperature. In the following calculation, N_{D}/N_{A} is set to be 5 to satisfy the condition for a n^{+}-p junction. N_{A} is then optimized at 300 K and 240 K, as show in _{0}A for PbSe homojunction at 300 K and 240 K are 2.5 × 10^{18} cm^{−3} and 1.5 × 10^{18}, respectively. For optimized carrier concentration the detector reaches BLIP limited D* at 210 K, 10 degrees higher than the “typical” carrier concentration.

At 240 K, the optimized N D = 1.5 × 10 18 cm − 3 ^{ }and N A = 3 × 10 18 cm − 3 are used to calculate maximum R_{0}A product in _{0}A value, the limitation D* is increased to 5.6 × 10^{10} HZ^{1/2}/W. At 300 K, the optimized N D = 2.5 × 10 18 and N A = 5 × 10 17 cm − 3 are applied to get the limitation D* which can be increased to 3 × 10^{10} HZ^{1/2}/W in

Performance limitation of the PbSe homojunction at high temperatures is theoretically studied in this paper. For such PV detectors thermal noise is dominating at temperatures higher than 240 K. BLIP limit could be achieved at temperatures around 210 K. The calculated peak D* are 3 × 10^{10} HZ^{1/2}/W and 5.6 × 10^{10} HZ^{1/2}/W at 300 K and 240 K respectively. Achieving D* of more than 10^{10} HZ^{1/2}/W in a MWIR detector at uncooled temperatures allows for its use in practical applications of high sensitivity.

The authors thank helpful discussions with Dr. Sung-Shik Yoo from Northrop Grumman. This work is partially supported by the DARPA WIRED program through Northrop Grumman, US army research office (ARO) under Grant No. W911NF-14-1-0312. The views, opinions and/or findings expressed are those of the author and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government.

Shi, X.H., Phan, Q., Weng, B.B., McDowell, L.L., Qiu, J.J., Cai, Z.H. and Shi, Z.S. (2018) Study on the Theoretical Limitation of the Mid-Infrared PbSe N^{+}-P Junction Detectors at High Operating Temperature. Detection, 6, 1-16. https://doi.org/10.4236/detection.2018.61001